This chapter investigates the convergence properties of the closed-loop system formed by an evolutionary dynamics model and a payoff dynamics model (EDM–PDM). The analysis combines the population game framework with the \(\delta \) -dissipativity tools introduced earlier. Sufficient conditions are rigorously established under which the equilibrium set of an EDM–PDM system coincides with the generalized Nash equilibrium set of the underlying population game and is asymptotically stable. These conditions certify the convergence of EDM–PDM dynamics, providing a reliable predictor of long-term strategic behavior in large populations. Leveraging \(\delta \) -dissipativity, convergence guarantees are derived for broad classes of interconnections between revision processes and payoff mechanisms. Representative EDM families and smoothing-anticipatory PDMs are analyzed to illustrate the results, showing that suitable conditions ensure convergence to a Nash equilibrium in unconstrained settings. The chapter concludes with simplified stability conditions and simulation-based validations that motivate the subsequent treatment of constrained cases.